Meir-Keeler Contraction In Rectangular M−Metric Space


 In this paper, we prove some fixed point theorems for a Meir-Keeler type Contraction in rectangular M−metric space. Thus, our results extend and improve very recent results in fixed point theory.


Introduction
Fixed point theory provides essential tools for solving problems arising in various branches of mathematical analysis, such as split feasibility problems, variational inequality problems, nonlinear optimization problems, equilibrium problems, complementarity problems, selection and matching problems, and problems of proving the existence of solution of integral and di erential equations. In Metric xed point theory, the wellknown Banach contraction principle [1] ensures the existence and uniqueness of xed points of contraction maps in the setting of complete metric spaces. There are two ways to generalized Banach contraction principle [1]. The rst one can change the active contraction and the second is to alter the underlying metric space. Many authors generalized the metric structure mainly: partial metric space [2], b−metric space [3], partial b−metric space [4], Branciari metric space [5,6], partial rectangular metric space [7], M−metric space [8], rectangular M−metric space [9], rectangular M b −metric space [10], extended rectangular Mr ξ −metric space [11], Mν−metric spaces [12] and some more [13]- [34].
In literature, there are many generalized contraction available. But due to their applicablity and impact, we record the few names as: Meir-Keeler contraction [35], Kannan contraction [36], Chatterjea [37], Boyd and Wong contraction [38] etc. In 2000, Branciari [6] introduced rectangular metric space and obtained certain xed point theorems. In 2014, Asadi et al. [8,39] introduced the M−metric space, which extends the p−metric space [2] and established many xed point theorems for Banach contraction principle and Meir-Keeler type contraction. In 2018, Özgur [9] extends both rectangular and M−metric space by introducing rectangular M−metric space and certain xed point theorems obtained therein. Meir-Keeler has a very signi cant place among the contraction condition. The de nition of Meir-Keeler contraction is as follows: De nition 1.1. [35] Let Mr be a non-empty set. A Meir-Keeler mapping is a mapping T : Mr → Mr on an rectangular M−metric space (X, Mr) such that Many authors studies Meir-Keeler contraction principle in di erent space. The aim of the paper is to study Meir-Keeler contraction in rectangular M−metric space.

Preliminaries
In this section, we collect some basic notions, de nitions, examples, lemmas and auxiliary results.
De nition 2.1. [6] If X be a non-empty set. A function r : X × X → R + is said to be a rectangular metric on X if it satis es the following (for all x, y ∈ X and for all distinct point u, v ∈ X \ {x, y}): Then, the pair (X, r) is called a rectangular metric space. Also, called Branciari distance space or generalized metric space [6].

De nition 2.2. [7]
If X be a non-empty set. A function ρ : X × X → R + is said to be a partial rectangular metric on X, if for any x, y ∈ X and for all distinct point u, v ∈ X \ {x, y} it satis es the following conditions: Then, the pair (X, ρ) is called a partial rectangular metric space. Notation 2.4 [9] The following notations are useful in the sequel: (i) mr xy := min{mr(x, x), mr(y, y)}, (ii) Mr xy := max{mr(x, x), mr(y, y)}.

De nition 2.4. [9]
If X be a non-empty set and mr : X × X → R + is a mapping. If it satisfying the following conditions for all x, y ∈ X: Then, the pair (X, mr) is called a rectangular M−metric space.
Notice that every m−metric is a rectangular m−metric. Example 2.1. [9] Let (X, d) be a rectangular metric space and a function ξ : [ , ∞) → [ , ∞) be a one-to-one and nondecreasing function with ξ ( ) = α such that Example 2.2. [9] Let (X, d) be a rectangular metric space and a function ξ : It is clear that each rectangular M−metric on X generates a T topology τm r on X. The set for all x ∈ X and ϵ > forms the base of τm r .
De nition 2.5. Proof. [9] Using the de nition of convergence and inequality (2.4), the proof of the condition (A) follows easily. From the Condition (mr xy ≤ mr(x, y)) and the Condition (A), we get limn→∞ min{mr(xn , xn), mr(x n− , x n− )} = limn→∞ mr xn x n− ≤ limn→∞ mr(xn , x n− ) = . Therefore, the Condition (B) holds. Since limn→∞ mr(xn , xn) = , the Condition (C) holds. Using the previous conditions and the de nition 2.5, we see that the Condition (D) holds. Then, T has a unique xed point u ∈ X, where mr(u, u) = .

Main results
The Following de nition is new version of the de nition in [35] for an Mr -Metric space.  Since by mr-completeness of X, there exists x * ∈ X such that xn → x * , i.e., lim n→∞ (mr(xn , x * ) − mr xn ,x * ) = .
Hence, x * = Tx * , that is, T has a xed point in X. Finally, we show the uniqueness of a xed point of T. Assume that T has two distinct xed points x * , x ** ∈ X such that Tx * = x * and Tx ** = x ** . Then, by the de nition of Meir-Keeler contraction if for all ϵ > o there exists δ > such that which is a contradiction. Hence, x * = x ** . This concludes the proof.
Corollary 3.1 [13] Let (X, d) be a complete metric space and let T be a continuous mapping from X into itself satisfying the following condition: for some K ∈ ( , ). Then, T has a unique xed point u ∈ X. Moreover, for all x ∈ X, the sequence {Tn(x)} converges to u.

Conclusion
As the rectangular m−metric is relatively new addition to the existing literature, therefore in this article, we established Meir-keeler contraction in rectangular M−metric space. As an application we derived some xed points of mappings of integral type.